## 3 January 2013

### Curie's principle and Lucretius' clinamen

In his 1894 paper [1], Pierre Curie noted:
1. The characteristic symmetry of a phenomenon is the maximum symmetry compatible with the existence of the phenomenon. A phenomenon can occur in a medium endowed with its characteristic symmetry or that of a subgroup of its characteristic symmetry.
2. To put it differently, certain symmetry elements can coexist with certain phenomena, but they are not needed. What is needed, is that certain symmetry elements be absent. It is the dissymmetry that engenders the phenomenon.
and proceeded to give a concise two-part statement of the symmetry principle:
1. When certain causes engender certain effects, the symmetry elements of the causes must also characterize the resulting effects.
2. When certain effects exhibit some dissymmetry, this dissymmetry must also characterize the engendering causes.
Curie was of course not the first to consider the symmetry of natural laws. In his paper he mentions explicitly (and makes use of) the cristallography treatises of his time. However, he is probably the first to have stated the symmetry principle at this level of generality. See [2,3] for a general discussion.

Spontaneous symmetry breaking seems to defy Curie's principle. The typical example is the buckling of a column under axial stress. Although initially the system has cylindrical symmetry, above a certain threshold the column shifts laterally, resulting in a lower-symmetry final state. Is this compatible with Curie's principle ?

One way out of this bind is to postulate the presence of a small static imperfection [4] or (in other contexts) of a quantum or thermal fluctuation. This asymmetry is irrelevant (and undetectable) under "normal" conditions and only has its effect at the transition, where the susceptibility of the system diverges. Thus, either the system was not symmetric enough to start with (due to the imperfection) or it cannot be (perfectly) symmetric due to the presence of the fluctuations. A second solution is to extend Curie's principle, so that it applies to ensembles, instead of individual systems [5]; more on this in a future post.

The first solution does raise a couple of interesting questions:
1. Far away from the transition, can we still consider the system as symmetric ?
2. How satisfying is this invocation of "invisible causes" ?
Here, however, I'm interested in whether this small deviation has something in common with Lucretius' concept of clinamen. As interpreted by Serres [6], the clinamen (declination, swerve) perturbs the primordial laminar flow, engendering a vortex (and, ultimately, the world).

It seems that the two concepts fulfill a similar role, by providing a primary and undetectable cause of some visible phenomenon.

In the case of the clinamen, these two properties open it to ridicule ([6], page 3):

From Cicero to Marx and beyond, down to us, the declination of atoms has been treated as a weakness of the atomic theory. The clinamen is an absurdity. A logical absurdity, since it is introduced without justification, the cause of itself before being the cause of all things; [...] an absurdity of physics in general, since experimentation cannot possibly reveal its existence.

Is this ridicule justified ? If so, how about the fluctuation that explains spontaneous symmetry breaking ?

[1] Pierre Curie, Sur la symétrie dans les phénomènes physiques, symétrie d'un champ électrique et d'un champ magnétique, Journal de Physique 3(1), 393-415, 1894.
[2] Jenann Ismael, Curie’s Principle, Synthese, 110, 167–190, 1997.
[3] Shaul Katzir, The emergence of the principle of symmetry in physics, Historical Studies in the Physical and Biological Sciences, 35(1), pages 35-65, 2004.
[4] Symmetry and Symmetry Breaking, in the Stanford Encyclopedia of Philosophy (section Spontaneous symmetry breaking.)
[5] Ian Stewart and Martin Golubitsky, Fearful Symmetry, Penguin Books, 1993.
[6] Michel Serres, The Birth of Physics, Clinamen Press, Manchester, 2000.